Using and Proving Angle Complements

Geometry

There are lots of relationships between angles that can be proven formally. For example, there are many situations where two seemingly unrelated angles can be shown to be complements of each other. Recall that two angles are complementary if the sum of their measures is 90º. For example, in Figure 9.4 there are several angles hanging around. There's ∠ABC, ∠CBD, and ∠DBE. Suppose that ∠CBD is a right angle. Then ∠ABC and ∠DBE are complementary. I'll write an informal proof of this.

Figure 9.4∠ABE is straight and ∠CBD is a right angle.

Example 4: Prove that if, as shown in Figure 9.4, ∠ABE is straight and ∠CBD is a right angle, then ∠ABC and ∠DEB are complementary.

Solution: All we need for an informal proof is a picture, the columns, and a game plan.

Here's the game plan: You are going to want to break apart your angles, so you know that the Angle Addition Postulate will be useful. You will need the definition of a straight angle, a right angle, and the definition of complementary angles in order to translate your angle characteristics into numbers. Then you'll be ready to use some algebra to bring it home.

Statements

Reasons

1.

∠ABE is straight and ∠CBD is a right angle

Given

2.

m∠ABE = 180º

Definition of straight angle

3.

m∠CBD = 90º

Definition of right angle

4.

m∠ABC + m∠CBD + m∠ABE

Angle Addition Postulate

5.

m∠ABC + m∠DBE = 180º

Substitution (steps 2, 3, 4)

6.

m∠ABC + m∠DBE = 90º

Algebra

7.

∠ABC and ∠DBE are complementary.

Definition of complementary angles

You started with your given information, used the definitions of the terms involved in the statement of the theorem, and finished up with what you wanted to prove. Every step has a reason that is either a definition, a postulate, an already-established theorem, or algebra. And the proof only required seven steps to write!

When I first introduced you to the concept of an angle, I threw out several angle relationships and gave brief explanations about why my claims were reasonable. Being the agreeable type of reader that you are, you didn't question me (or if you did, I didn't hear you). I'll take a minute and address one of the statements I made that might have raised an eyebrow or two. The statement in question is “the complement of an acute angle is an acute angle.” There's no time like the present to write a formal proof of this potentially bold claim.

Example 5: Write a formal proof that the complement of an acute angle is an acute angle.

Solution: Let's try a new approach. Let's work through the five steps in writing a proof. (Note the sarcasm.)

1. State the theorem.

Theorem 9.4: The complement of an acute angle is an acute angle.

2. Draw a picture. You need two angles, one of which is acute, whose measures add up to 90º. In other words, the two angles must combine to form a right angle. I have drawn these two angles in Figure 9.5.

Figure 9.5An acute angle ∠ABC and its complement, ∠CBD.

3. Interpret what is given in terms of your picture. You are given an acute angle ∠ABC, and its complement ∠CBD.

4. Interpret what you want to prove in terms of your drawing. You want to prove that ∠CBD is acute.

5. Write the proof. Again, you need a game plan. You are given an acute angle, so the definition of an acute angle will be useful. Because you are combining angles, you might want to use the Angle Addition Postulate. Because you are trying to show that ∠CBD is acute, you need to show that m∠CBD < 90º. So you'll need to refresh what it means for m∠CBD to be less than 90º. Recall that x is less than y if there exists a positive number z for which x + y = z. Because the measure of an angle is a positive number between 0 and 180 (recall the Protractor Postulate), you should be all set.

Statements

Reasons

1.

∠ABC is acute, and ∠ABC and ∠CBD are complementary.

Given

2.

m∠ABC + m∠CBD = 90º

Definition of complementary angles

3.

m∠ABC > 0

Protractor Postulate

4.

m∠CBD < 90º

Definition of inequality

5.

∠CBD is acute

Definition of acute angle

Is it my imagination, or are these proofs getting shorter and easier to write?